The Cyclic and Epicyclic Sites

The Cyclic and Epicyclic Sites

ALAINCONNES(*) - CATERINACONSANI(**)(***)

ABSTRACT- We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in char- acteristic one over algebraic extensions of the infinite semifield of ``max-plus integers'' Zmax. An object of this category is a pair (E;K) of a semimoduleEover an algebraic extensionKofZmax. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic toposdNof [6] and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 18B25; 20L05, 19D55.

KEYWORDS. Grothendieck topos, Cyclic category, Groupoids, Characteristic one, Projective geometry.

1. Introduction

The theory of topoi of Grothendieck provides the best geometric framework to understand cyclic homology and thel-operations using the topos associated to the cyclic category [3] and its epicyclic refinement [5]. Given a topos T a basic question is to determine the category of points of T, i:e: of geometric morphisms from the topos of sets toT. In this paper we show how to describe the category of points of the epicyclic topos in terms of projective geometry in characteristic 1. Given a small category C, we denote by C^ the topos of con- travariant functors from C to the category Gets of sets. The epicyclic topos …L^op†^{^} is obtained by taking the opposite of the epicyclic category L. This

(*) Indirizzo dell'A.: ColleÁge de France, 3 rue d'Ulm, Paris F-75005 France, I.H.E.S. and Ohio State University.

E-mail: alain@connes.org

(**) Indirizzo dell'A.: Department of Mathematics, The Johns Hopkins University, Baltimore MD 21218 USA.

E-mail: kc@math.jhu.edu

(***) C. Consani would like to thank the ColleÁge de France for some financial support.

choice is dictated by the following natural construction. A commutative ring R determines a covariant functor ( )^\:Fin !Ab from the category of finite sets to that of abelian groups. This functor assigns to a finite set J the tensor powerR^JˆN

j2JR. As explained in geometric terms here below, there is also a natural covariant functor L !Fin. The composite of these two functors L !Abprovides, for any commutative ringR, acovariantfunctorR^\from the epicyclic category to the category of abelian groups. In geometric termsR^\is a sheaf of abelian groups over the topos …L^op†^{^}. Both the cyclic homology of R and its l-operations are completely encoded by the associated sheafR^\. In [5], we provided a conceptual understanding of the epicyclic category as projective geometry over the semifieldF:ˆZmax of the tropical integers. In these terms the functor ( )^\ considered above assigns to a projective space the underlying finite set. This article pursues the relation between the epicyclic topos and (projective) geometry in characteristic 1 in more details. Our main result is the following (cf.Theorem 4.1)

THEOREM. The category of points of the epicyclic topos…L^op†^{^}is equivalent to the categoryPwhose objects are pairs(K;E), where K is an algebraic extension of FˆZ_maxand E is an archimedean semimodule over K. The morphisms inPare projective classes of semilinear maps and injective semifield morphisms.

It is important to realize the relevance of the language of Grothendieck topoi to interpret, for instance, the action of the edgewise subdivision on the points of the simplicial toposD. The preliminary Section 2 is dedicated to this description. It is^ well known (cf.[10]) that the points ofD^correspond to intervals,i:e:totally ordered setsIwith a smallest elementband a largest elementt6ˆb. For each integerk>0, the edgewise subdivision Sdk defines an endofunctor of the simplicial categoryD and one obtains in this way an action of the monoõÈdNby geometric morphisms on the toposD. We show that the action of the edgewise subdivision on the points of^ D^ is given by the operation of concatenation ofkcopies of the intervalI: the inter- mediate top pointtj of the copyIj is identified with the bottom point bj‡1 of the subsequent copyIj‡1. Then, we form the small categoryD^opjNcrossed product ofD^opby the transposed action Sdof N(i.e.Sd_k(f)ˆSd_k(f), wheref7!fis the anti-isomorphismD !D^op).

Section 3 gives a description of the epicyclic category in terms of oriented groupoõÈds. The ambiguõÈty in the choice of a representative of a projective class of semilinear maps in the category P of Theorem 4.1 is inconvenient when working, for example, with colimits. In Section 3.3 we provide a description of the cyclic and the epicyclic categories in terms of a category g of oriented groupoõÈdswhose morphisms are no longer given by equivalence classes. There are by now a number of equivalent descriptions of the cyclic and epicyclic cat- egories, ranging from the most concrete i:e: given in terms of generators and relations, to the most conceptual as in [5]. The description of these categories in

terms of oriented groupoõÈds turns out to be very useful to determine the points of the epicyclic topos by considering filtering colimits, in the categoryg, of the special points provided by the Yoneda embedding of the categories. It is in fact well known that any point of a topos of the form C^ is obtained as a filtering colimit, in the category of flat functors C !Gets, of these special points. On the other hand, there is no guarantee ``a priori'' that this colimit process yields the same result as the colimit taken in the category g. This matter is solved in two steps and in concrete terms in Section 4. In Proposition 4.3 we show how to associate to a pair (K;E) as in the above Theorem a point of…L^op†^{^}. Conversely, in §§4.2-4.4 we explain a geometric procedure that allows one to reconstruct the structure of an oriented groupoõÈd from the flat functor naturally associated to a point of…L^op†^{^}.

In §4.8 we explore the relations of …L^op†^{^} with the arithmetic site dN, as re- cently defined in [6]. LetNbe the small category with a single objectand whose endomorphisms End()ˆN form the multiplicative semigroupN of positive integers. One has a canonical functor Mod:L^op !N which is trivial on the objects and associates to a semilinear map of semimodules over FˆZmax the corresponding injective endomorphism Fr_n2End(F) (cf. [5] for details). This functor induces a geometric morphism of topoi Mod:…L^op†^{^} !dN. The sub- category of L^op which is the kernel of this morphism is the cyclic category L (L'L^op).

In Appendix A we view the l-operations as elements L^k_n of the convolution ring Z[D^opj N] with integral coefficients. We review their geometric meaning and the geometric proof of their commutation (cf. [9]) with the Hochschild boundary operator.

Appendix Bis dedicated to the description of the cyclic homology of cyclic modules (cf.[3]) and its extension to epicyclic modules [8]. We stress the nuance betweenLandL^op in a hopefully clear form. AnepicyclicmoduleEis acovariant functorL !Ab. These modules correspond to sheaves of abelian groups on the topos…L^op†^{^}. At this point the nuance between the epicyclic category and its dual plays an important role since unlike the cyclic category the epicyclic category is not anti-isomorphic to itself. As explained earlier on in this introduction, a commu- tative ringRgives rise naturally to an epicyclic moduleR^\and it is well known (cf.

[8]) that thel-operations on cyclic homology ofRare obtained directly through the associated epicyclic module. We provide a simple and conceptual proof of the commutation of thel-operations with theBoperator of cyclic theory. Finally, we point out that the extended framework of epicyclic modules involves many more modules than those arising by composition, as explained earlier, from a covariant functor Fin !Ab. In fact, these particular (epicyclic) modules have integral weights and thel-operations decompose their cyclic homology as direct sums of modules on whichL^k_n acts by an integral power ofk. This integrality property no longer holds for general epicyclic modules as can be easily checked by applying a twisting argument.

2. The action of the edgewise subdivision on points ofD^

We recall that the simplicial categoryDis the small category with objects the totally ordered sets [n]:ˆ f0;. . .;ng, for each integern0, and morphisms non- decreasing maps.

In this section we study, using the formalism of topoi, the edgewise subdivision functors Sd_k :D !D, fork2Nand their action on the points of the simplicial toposD.^

2.1 ±The edgewise subdivision functorsSd_k

LetFbe a finite, totally ordered set andk2Na positive integer. We define the set

Sd_k(F):ˆ f0;. . .;k 1g F …1†

to be the cartesian product of the finite ordered setf0;. . .;k 1gwithF, endowed with the lexicographic ordering. For f 2HomD(F;F⁰) a non-decreasing map (of finite, totally ordered sets), we let

Sd_k(f):ˆIdf :Sd_k(F)!Sd_k(F⁰) …2†

PROPOSITION 2.1. For each k2N, (1) and (2) define an endofunctor Sd_k:D !D. They fulfill the property

Sd_kk⁰ˆSd_kSd_k⁰; 8k;k⁰2N:

PROOF. The totally ordered sets Sd_k([n]) and [k(n‡1) 1] have the same cardinality and are canonically isomorphic. The unique increasing bijection Sd_k([n])![k(n‡1) 1] is given by

(a;i)7!i‡a(n‡1); 8a2 f0;. . .;k 1g; i2 f0;. . .;ng:

Let f 2HomD([n];[m]) then by definition Sd_k(f)2HomD(Sd_k([n]);Sd_k([m])) is given by

Sd_k(f)(i‡a(n‡1))ˆf(i)‡a(m‡1); 8i;a; 0in; 0ak 1:

…3†

One checks directly that Sdkk⁰ ˆSdkSdk⁰. p

We transfer the functors Sdk to the opposite category D^op of finite intervals.

Recall that by definition, an interval I is a totally ordered set with a smallest elementband a largest elementt6ˆb. The morphisms between intervals are the non-decreasing maps respectingbandt,i:e:f :I!J,f(b_I)ˆb_J,f(t_I)ˆt_J.

For all n0 we denote by n:ˆ f0;. . .;n‡1g. The interval n para- metrizes the hereditary subsets of [n]: indeed, to j2n corresponds [j;n]:ˆ

fx2[n]jxjg, the latter set is empty forjˆn‡1. The duality betweenDand D^opis then provided by the contravariant functorD ! D^op, [n]7!n, which acts

on morphisms as follows

Hom_D([n];[m])3f !f2Hom_D^op(m;n); f ¹([j;m])ˆ[f(j);n]; 8j2m: …4†

LetIbe an interval, andk2N, then one lets Sd_k(I) to be the quotient of the totally ordered setf0;. . .;k 1g IˆSd_k(I) (with lexicographic ordering) by the equiva- lence relation (j;t_I)(j‡1;b_I) forj2 f0;. . .;k 2g. This defines an endofunctor Sd_k of the category of intervals whose action on morphisms sends f :I!J to Sd_k(f)ˆIdf. By restriction to finite intervals one obtains an endofunctor Sd_k:D^op !D^op.

In particular, the interval Sd_k(n) has k(n‡2) (k 1)ˆk(n‡1)‡1 ele- ments and one obtains a canonical identification of Sd_k(n) with the hereditary subsets of Sd_k([n]) as follows

f0;. . .;k 1gn3(b;j)7! f(a;i)2 f0;. . .;k 1g [n]ja>boraˆb& jig:

Note that the right hand side of the above formula depends only upon the class of (b;j)2Sd_k(n).

LEMMA2.2. For f 2HomD([n];[m]), one has(Sdk(f))ˆSd_k(f):

PROOF. The morphism (Sd_k(f)) is defined by the equivalence Sd_k(f)(x)y()(Sd_k(f))(y)x

Let xˆi‡a(n‡1), yˆj‡b(m‡1) with 0in, 0jm, 0ak 1, 0bk 1. Then by (3) one has Sd_k(f)(x)ˆf(i)‡a(m‡1), thus the condition Sdk(f)(x)ydetermines (Sdk(f))(y) as follows

f(i)‡a(m‡1)j‡b(m‡1)()a>boraˆbandf(i)j ()i‡a(n‡1)f(j)‡b(n‡1)ˆSd_k(f)(y):

This provides the required equality (Sdk(f))ˆSd_k(f). p THEOREM2.3. The actionSddkof the geometric morphismSdk (k2N) on the points of the toposD^is described by the endofunctorSd_kon the category of intervals.

PROOF. One can prove this theorem using the fact that any point ofD^is obtained as a filtering colimit of the points associated to the Yoneda embedding ofD^opin the category of points ofD. One shows that on such points the action^ Sdc_kcoincides with the functor Sd_k:D^op !D^op. We shall nevertheless find it more instructive to give, in §2.3, a concrete direct proof of the equality between the following two flat functorsD !Getsassociated to an intervalI

…5† F1…‰nŠ † ˆHom_D^op…n;Sd_k…I††;F2…‰nŠ † ˆ

ˆ a

m0

…Hom…m;I† Hom_D^op…n;Sd_k…m†††

!

=

hereF2is the inverse image functor of the pointpIofD^applied to the contravariant functorY:D !Gets,YˆXSd_k, whereXˆh_[n] is the Yoneda embedding, so that

Y([m])ˆX(Sd_k([m]))ˆHom_D(Sd_k([m]);[n])ˆHom_D^op(n;Sd_k(m)):

…6†

Thus F2 corresponds to the point Sddk(pI) and the equality between F1 and F2

(cf.Lemma 2.7) yields the result. p

COROLLARY2.4.The point of the simplicial toposD^ associated to the interval [0;1]Ris a fixed point for the action of NonD^.

PROOF. The statement follows using the affine isomorphism Sd_k([0;1])ˆ f0;. . .;k 1g [0;1]![0;1]; (a;x)7!a

k‡x k: …7†

2.2 ±Canonical decomposition ofW2Hom_D^op(n;Sd_k(I))

LetIbe an interval,I=be the quotient ofIby the identificationbt. Con- sider the map p:Sd_k(I)!I=, (j;x)7!x. For W2Hom_D^op(n;Sd_k(I)), we define the rank of W as the cardinality of the set ZˆI^o\Range(pW), where I^oˆIn fb;tg.

PROPOSITION 2.5. LetW2Hom_D^op(n;Sd_k(I))and r its rank. Then, one has a unique decomposition

WˆSd_k(a)b; b2Hom_D^op(n;Sd_k(r)); a2Hom_D^op(r;I):

Moreover, the morphism a2Hom_D^op(r;I) is the unique increasing injection f1;. . .;rg,!I^owhich admits Z as range. The compositeprbis surjective, where p_r:Sd_k(r)!r=is the canonical surjection.

PROOF. Leta2Hom_D^op(r;I) be the map whose restriction tof1;. . .;rgis the unique increasing injection intoI^owhich admitsZas range. Recall that an element x2Sd_k(I) is given by a pairxˆ(j;y)2 f0;. . .;k 1g IˆSd_k(I) with the iden- tifications (j;t)(j‡1;b) forj2 f0;. . .;k 2g. Similarly an elementz2Sd_k(r) is given by a pair zˆ(i;u)2 f0;. . .;k 1g rˆSd_k(r) with the identifications (j;r‡1)(j‡1;0) for j2 f0;. . .;k 2g. Let s2n ˆ f0;. . .;n‡1g, then W(s)2Sd_k(I) is given by a pairW(s)ˆ(j;y)2 f0;. . .;k 1g IˆSdk(I) unique up to the above identifications. Ify2 fb;tg, one definesb(s)ˆ(j;0)2Sd_k(r) ifyˆb, and b(s):ˆ(j;r‡1)2Sd_k(r) if yˆt. This definition is compatible with the identifications. Let us now assume that y2 fb;= tg. Then y2Zand there exists a unique elementv2 f1;. . .;rgsuch thatyˆa(v). One then definesb(s):ˆ(j;v)2

Sd_k(r). The mapb:n!Sd_k(r) so defined is non-decreasing,i.e.fors<s⁰one hasb(s⁰)b(s) since the inequalityW(s⁰)W(s) shows that eitherj⁰>jin which case (j⁰;v⁰)(j;v) is automatic, orjˆj⁰and in that casey⁰>ywhich shows thatv⁰v.

Moreover since Sd_k(a)ˆIdaone hasWˆSd_k(a)b.

We prove the uniqueness of this decomposition. Since W2Hom_D^op(n;Sd_k(I)) preserves the base points, Range (pW) contains the base point and its cardinality is r‡1. Thus the map a2Hom_D^op(r;I) is the unique map whose restriction to f1;. . .;rgis the increasing injection toI^oand which admitsZˆI^o\Range (pW) as range. Moreover a is injective and so is Sd_k(a). Thus the map b2 Hom_D^op(n;Sd_k(r)) is uniquely determined by the equalityWˆSd_k(a)b. Finally prb:n!r=is surjective since otherwise the range of Sd_k(a)b would be

strictly smaller than the range ofW. p

COROLLARY2.6.For any interval Ithe map

Hom_D^op(n;I)Hom_D^op(n;Sd_k(n))!Hom_D^op(n;Sd_k(I));(a;b)7!Sd_k(a)b is surjective.

2.3 ±Explicit description of the isomorphism F₁'F₂

LetF_j:D !Getsbe the flat functors defined in (5). By definition F2([n])ˆ a

m0

(Hom(m;I)Hom_D^op(n;Sd_k(m)))

!

= …8†

where the equivalence relation is generated by (af;b)(a;Sd_k(f)b)

forf 2Hom_D^op(m;r),b2Hom_D^op(n;Sd_k(m)),a2Hom(r;I).

LEMMA2.7. The map

F:F2([n])!F1([n])ˆHom_D^op(n;Sd_k(I)); (a;b)7!Sd_k(a)b is a bijection of sets.

PROOF. The mapFis well defined sinceF(af;b)ˆF(a;Sd_k(f)b). Corollary 2.6 shows thatFis surjective. To show the injectivity it is enough to prove that for any (a;b)2Hom(m;I)Hom_D^op(n;Sd_k(m)) one has (a;b)(a_c;b_c) where

WˆSd_k(a_c)b_c; b_c2Hom_D^op(n;Sd_k(r)); a_c 2Hom_D^op(r;I) is the canonical decomposition ofWˆSd_k(a)b.

One has the canonical decompositionbˆSd_k(a0)b₀withb₀2Hom_D^op(n;Sd_k(`)), Id`b₀surjective. Thus (a;b)(aa0;b₀). Since Id`b₀is surjective, Rangeaa0

Range IdW= Rangeac, and thusaa0ˆacr

(a;b)(aa0;b0)ˆ(acr;b0)(ac;Sd_k(r)b0)ˆ(ac;bc):

p

2.4 ±The small categoryD^opj N

We denote by D^opj N the small category semi-direct product ofD^opby the action of N implemented by the endofunctors Sd_k, fork2N. It has the same objects as D^op while one adjoins to the collection of morphisms of D^op the new morphismsp^k_n:Sd_k(n)ˆ…k(n‡1) 1†!n such that

p^k_np^{`</sup><sub>k(n‡1) 1</sub>ˆp<sup>k`}_n 2Hom_D^op _N…(k`(n‡1) 1);n† …9†

wherep^k_nimplements the endofunctor Sd_k,i.e.

ap^k_nˆp^k_mSd_k(a); 8a2Hom_D^op(n;m):

…10†

Using this set-up one checks that any morphismfinD^opj N is uniquely of the formfˆp^k_nawithaa morphism inD^op. Any suchfcomposes as follows

(p^k_mb)(p^{`</sup><sub>n</sub>a)ˆp<sup>k`}_m(Sd_`(b)a) …11†

wherea2Hom_D^op(r; `(n… ‡1) 1†) and

Sd<sub>`</sub>(b)2Hom<sub>D</sub><sup>op</sup>(…`(n‡1) 1†;…k`(m‡1) 1†)

so that Sd<sub>`</sub>(b)amakes sense and belongs to Hom<sub>D</sub><sup>op</sup>(r;…k`(m‡1) 1†). Using Proposition 2.1 one checks that, if one takes (11) as a definition, the product is as- sociative.

LetFinbe the category of finite pointed sets and letF be the functor which associates to an interval Ithe pointed set IˆI= with base point the class of bt. To any morphism of intervals f :I!J corresponds the quotient map f which preserves the base point. By restrictingFtoD^opone gets a covariant functor F :D^op !Fin. The following Proposition shows that F can be extended to D^opj N.

PROPOSITION 2.8. For any n0;k2N, let (p^k_n):F(Sd_k(n))! F(n) be given by the residue modulo n‡1. Then the extension of the functor F on morphisms given by

fˆp^k_na7!f :ˆ(p^k_n)a determines a functorF :D^opj N !Fin.

PROOF. One checks directly that the definition of (p^k_n) is compatible with the rules (9) and (10) so that the required functoriality follows. p

3. The epicyclic category and the oriented groupoõÈds

3.1 ±Generalities on groupoõÈds

A groupoõÈdGis a small category where the morphisms are invertible. Given a subset XG of a groupoõÈd, we set X ¹:ˆ fg ¹jg2Xg. Let G⁽⁰⁾ be the set of objects ofGand denote byr;s:G!G⁽⁰⁾the range and the source maps respec- tively. We viewG⁽⁰⁾as the subset of units ofG. The following definition is a direct generalization to groupoõÈds of the notion of right ordered group (cf. [7])

DEFINITION3.1. An oriented groupoõÈd(G;G_‡)is a groupoõÈd G endowed with a subcategory G‡G, such that the following relations hold

G_‡\G_‡¹ˆG⁽⁰⁾; G_‡[G_‡¹ˆG:

…12†

Let (G;G‡) be an oriented groupoõÈd and let x2G⁽⁰⁾. The set Gx:ˆ

fg2Gjs(g)ˆxgis endowed with the total order defined by gg⁰()g⁰g ¹2G‡:

…13†

This order is right invariant by construction: i.e. for any b2G, with r(b)ˆx, one has

gg⁰()gbg⁰b:

In the following subsections we describe two constructions of oriented groupoõÈds associated to a group action.

3.1.1 ± GˆX^j H.

LetHbe a group acting on a setX. Then the semi-direct productG:ˆX^j H is a groupoõÈd with source, range and composition law defined respectively as follows

s(x;h):ˆx; r(x;h):ˆhx; (x;h)(y;k):ˆ(y;hk):

(As in any groupoõÈd the compositiongg⁰is only defined whens(g)ˆr(g⁰) which holds here if and only ifxˆky). One has a canonical homomorphism of groupoõÈds r:G!H,r(x;h)ˆh.

LEMMA3.2. Let(H;H‡)be a right ordered group. Assume that H acts on a set X. Then the semi-direct product GˆX^j H with G_‡:ˆr ¹(H_‡) is an oriented groupoõÈd .

PROOF. By definition, the subsetH‡Hof the groupHis stable under product and fulfills the equalities:H‡\H_‡¹ˆ f1g; H‡[H_‡¹ˆH. This implies (12) using

r ¹(f1g)ˆG⁽⁰⁾. p

Let, in particular, (H;H‡)ˆ(Z;Z‡) act by translation on the set XˆZ=(m‡1)Z of integers modulo m‡1. Then one obtains the oriented groupoõÈd

g(m):ˆ(Z=(m‡1)Z)^j Z:

…14†

The oriented groupoõÈdsg(m) will play a crucial role in this article.

3.1.2 ± Gˆ(XX)=H.

Let Hbe a group acting freely on a set X. LetG(X;H)ˆ(XX)=Hbe the quotient ofXXby the diagonal action ofH

G(X;H):ˆ(XX)= (x;y)(h(x);h(y)); 8h2H:

Letrandsbe the two projections ofG(X;H) onG⁽⁰⁾:ˆX=Hdefined byr(x;y)ˆx ands(x;y)ˆy. Letg;g⁰2G(X;H) be such thats(g)ˆr(g⁰). Then, forg(x;y) and g⁰(x⁰;y⁰), there exists a unique h2H satisfying x⁰ˆh(y): this because s(g)ˆr(g⁰) andHacts freely onX. Then, the pair (h(x);y⁰) defines an element of G(X;H) independent of the choice of the pairs representing the elementsgandg⁰. We denote by gg⁰ the class of (h(x);y⁰) inG(X;H). This construction defines a groupoõÈd law onG(X;H)ˆ(XX)=H.

LEMMA3.3. Let H be a group acting freely on a set X. Assume that X is totally ordered and that H acts by order automorphisms. Then G(X;H)ˆ(XX)=H is an oriented groupoõÈd with

G‡(X;H)ˆ f(x;y)2G(X;H)jxyg:

…15†

PROOF. Since H acts by order automorphisms the condition xy is in- dependent of the choice of a representative (x;y) of a given g2G(X;H)ˆ (XX)=H. This condition defines a subcategoryG_‡(X;H) ofG(X;H). The condi- tions (12) then follow sinceXis totally ordered. p LEMMA3.4. Let XˆZwith the usual total order. Let m2Nand let the group Z act on X by h(x):ˆx‡(m‡1)h, 8x2X;h2Z. Then the oriented groupoõÈd Gˆ(XX)=Zis canonically isomorphic to the oriented groupoõÈdg(m)of(14).

PROOF. The associated oriented groupoõÈd (G;G_‡) is by construction the quo- tient of ZZ by the equivalence relation: (x;y)(x‡`(m‡1);y‡`(m‡1)), 8`2Z. Thus the following map defines a bijective homomorphism of groupoõÈds

c:G!g(m)ˆ(Z=(m‡1)Z)^j Z; c(x;y)ˆ(p(y);x y)

where p:Z!Z=(m‡1)Z is the natural projection. One has by restriction G_‡!^c

g_‡(m), since xy()x y0, so that c is in fact an isomorphism of

oriented groupoõÈds. p

3.2 ±The oriented groupoõÈd associated to an archimedean set

In this section we explain how to associate an oriented groupoõÈd to an archi- medean set and describe the special properties of the oriented groupoõÈds thus obtained. We first recall from [4] the definition of an archimedean set.

DEFINITION3.5. An archimedean set is a pair(X;u)of a non-empty, totally ordered set X and an order automorphismu2AutX, such thatu(x)>x,8x2X.

The automorphism u is also required to fulfill the following archimedean property

8x;y2X; 9n2Ns:t: yuⁿ(x):

Let (X;u) be an archimedean set and letG(X;u) be the oriented groupoõÈd as- sociated by Lemma 3.3 to the action ofZonXby integral powers ofu. Thus

G(X;u):ˆ(XX)=; (x;y)(uⁿ(x);uⁿ(y)); 8n2Z and

G_‡(X;u):ˆ f(x;y)2G(X;u)jxyg:

…16†

Next proposition describes the properties of the pair (G(X;u);G‡(X;u)) so obtained.

PROPOSITION 3.6. The oriented groupoõÈd (G;G_‡)ˆ(G(X;u);G_‡(X;u)) fulfills the following conditions

(1) 8x;y2G⁽⁰⁾,9g2G_‡ s.t. s(g)ˆy,r(g)ˆx.

(2) Forx2X, the ordered groupsG^x_x:ˆ fgjs(g)ˆr(g)ˆxgare isomorphic to(Z;).

(3) Let g2G with s(g)ˆy and r(g)ˆx. Then the map: G^y_y3r7!g rg ¹2G^x_xis an isomorphism of ordered groups.

PROOF. Sinceuis an order automorphism ofX, the groupZacts by order au- tomorphisms. We check the three conditions (i)-(iii).

(i) For x;y2X, there exists n2N such that uⁿ(x)y. Then g(uⁿ(x);y) belongs toG_‡ands(g)ˆy,r(g)ˆx.

(ii) Letx2X. The conditionss(g)ˆr(g)ˆximply that the class ofg2G(X;u) admits a unique representative of the form (uⁿ(x);x). One easily checks that the mapG^x_x!(Z;), (uⁿ(x);x)7!nis an isomorphism of ordered groups.

(iii) Let x;y2X with g(x;y). Then for r(uⁿ(y);y)2G^yy one gets grg ¹(uⁿ(x);x), thus the unique isomorphism with (Z;) is preserved. p Let (G;G_‡) be an oriented groupoõÈd fulfilling the three conditions of Propo- sition 3.6. Let x2G⁽⁰⁾, consider the set Gxˆ fg2Gjs(g)ˆxg with the total order defined by (13) and with the action of Z given, for g_x2G‡ the positive

generator of G^x_x, by

u(g):ˆgg_x: …17†

When one applies this construction to the case (G;G_‡)ˆ(G(X;u);G(X;u)_‡), for (X;u) an archimedean set, withx2G⁽⁰⁾ˆX=uone obtains, after choosing a lift x~2Xofx, an isomorphism

j~x:X! G_x; j~x(z)ˆ(z;x):

…18†

The following proposition shows that the two constructions (X;u)7!G(X;u) and (G;G‡)7!(Gx;u) are reciprocal.

PROPOSITION3.7. Let(G;G_‡)be an oriented groupoõÈd fulfilling the conditions of Proposition3.6 and let x2G⁽⁰⁾. Consider the set Gx:ˆ fg2Gjs(g)ˆxg ˆX endowed with the total order(13)and the action ofZon it given by(17). Then(X;u) is an archimedean set and one has an isomorphism of groupoõÈds

(G(X;u);G(X;u)_‡)(G;G_‡):

PROOF. The implicationgg⁰ˆ)u(g)u(g⁰) follows since right multiplication preserves the order. Moreover, the condition (iii) of Proposition 3.6 implies that u(g)>g,8g2XˆG_x. Next we show that the archimedean property holds on (X;u).

Letgg⁰2G_x, withyˆr(g) andy⁰ˆr(g⁰). By applying the condition (i) of Pro- position 3.6, we choose d2G‡ such that s(d)ˆy⁰ and r(d)ˆy. Theng⁰⁰ˆdg⁰ fulfillss(g⁰⁰)ˆs(g) andr(g⁰⁰)ˆr(g) and thus there existsn2Zsuch thatg⁰⁰ˆggⁿ_x. Moreover, one hasg⁰⁰ˆdg⁰g⁰. It follows that (X;u) is an archimedean set. If one replaces x2G⁽⁰⁾ by y2G⁽⁰⁾, then the condition (i) implies that there exists a2G_‡ with s(a)ˆy,r(a)ˆx. Then the map G_x!G_y,g7!ga is an order iso- morphism which satisfies

(gg_x)aˆ(ga)(a ¹g_xa):

Since condition (iii) impliesa ¹g_xaˆg_y, one obtains an isomorphism of the corresponding archimedean sets.

Finally, we compare the pair (G;G‡) with (G(X;u);G‡(X;u)). We define a map f :G(X;u)!G as follows: given a pair (g;g⁰) of elements of XˆG_x, one sets f(g;g⁰):ˆgg^{0 1}. One has

f(u(g);u(g⁰))ˆf(gg_x;g⁰g_x)ˆgg^{0 1}ˆf(g;g⁰):

To show that f is a groupoõÈd homomorphism it is enough to check that f(g;g⁰)f(g⁰;g⁰⁰)ˆf(g;g⁰⁰) and this can be easily verified. Next we prove thatf is bijective. Leta2G. By applying condition (i) of Proposition 3.6, there existsg2G such that r(g)ˆr(a) and s(g)ˆx. Let then g⁰ˆa ¹g. Since s(g⁰)ˆx, both g;g⁰ belong toXˆGxand moreoverf(g;g⁰)ˆashowing thatf is surjective. Letg_j;g_j⁰ be elements of XˆGx such that f(g₁;g₁⁰)ˆf(g₂;g₂⁰). One then has g₁g₁^{0 1}ˆg₂g₂^{0 1} and henceg₂¹g₁ˆg₂^{0 1}g₂ˆgⁿ_x for somen2Z. It follows

that (g₁;g₁⁰)ˆ(uⁿ(g₂);uⁿ(g₂⁰)) which shows thatf is also injective. Finally, for any g;g⁰2XˆG_x one hasgg⁰()gg^{0 1}2G_‡ showing thatf, so defined, is an

order isomorphism. p

3.3 ±The category of archimedean sets in terms of oriented groupoõÈds

In this section we extend the above construction (X;u)!G(X;u) of the ori- ented groupoõÈd associated to an archimedean set to a functor G connecting the categoryArc^j N of archimedean sets to that of oriented groupoõÈds. We recall, from [4], the definition of the category of archimedean sets.

DEFINITION3.8. The objects of the categoryArc^j Nare the archimedean sets (X;u) as in Definition 3.5, the morphisms f :(X;u)!(X⁰;u⁰) in Arc^j N are equivalence classes of maps

f :X!X⁰; f(x)f(y) 8xy; 9k>0; f(u(x))ˆu^0k(f(x)); 8x2X …19†

where the equivalence relation identifies two such maps f and g if there exists an integer m2Zsuch that g(x)ˆu^0m(f(x)),8x2X.

Given a morphism f :(X;u)!(X⁰;u⁰) in Arc^j N as in (19), the map G(f) sending (x;y)7!(f(x);f(y)) is well defined since

XX3(x;y)(x⁰;y⁰)ˆ)(f(x);f(y))(f(x⁰);f(y⁰)):

Moreover G(f):G(X;u)!G(X⁰;u⁰) does not change if we replace f by g(x)ˆ u^0m(f(x)) since this variation does not alter the element

(f(x);f(y))(g(x);g(y))2G(X⁰;u⁰):

PROPOSITION3.9. The association(X;u)7!G(X;u), f7!G(f)defines a faithful functor G from Arc^j N to the category of oriented groupoõÈds. For any non- trivial morphism of oriented groupoõÈds r:G(X;u)!G(X⁰;u⁰) there exists a (unique) morphism

f 2Hom_{Arc N}((X;u);(X⁰;u⁰)) s:t: rˆG(f):

PROOF. By construction the mapG(f) is a morphism of groupoõÈds since G(f) (x;… y)(y;z)† ˆ(f(x);f(z))ˆG(f)((x;y))G(f)((y;z)):

Moreover it is a morphism of oriented groupoõÈds since by (19), one hasf(x)f(y), 8xy so that

(x;y)2G_‡(X;u)ˆ)(f(x);f(y))2G_‡(X;u):

We show that the functorGis faithful. Assumef;g2Hom_{Arc N}((X;u);(X⁰;u⁰)) are such that G(f)ˆG(g). Then for any (x;y)2XX there exists an integer

nˆn(x;y)2Zsuch that

g(x)ˆu^0n(x;y)(f(x)); g(y)ˆu^0n(x;y)(f(y)):

Since u⁰ acts freely onX⁰, the integern(x;y) is unique. The first and the second equations prove thatn(x;y) is independent ofyandxrespectively. Thus one derives thatfandgare in the same equivalence class,i.e.they define the same element of Hom_{Arc N}((X;u);(X⁰;u⁰)).

Let nowr:G(X;u)!G(X⁰;u⁰) be a non-trivial morphism of oriented groupoõÈds.

Letx2G⁽⁰⁾,yˆr(x): the ordered group morphismr:G^x_x!G^0y_y is non-constant and is given byr(g_x)ˆg^k_y, for somek>0. The mapr:Gx!G⁰_yis non-decreasing since

gg⁰ˆ)g⁰g ¹2G‡ˆ)r(g⁰)r(g) ¹2G⁰_‡:

Given an archimedean set (X;u) and an element z2X, we claim that the map c_X;z:X!G(X;u) defined byc_X;z(y):ˆ(y;z) is an order preserving bijection ofX with G(X;u)_x, wherexis the class ofzinX=u. Indeed, every element ofG(X;u)_x admits a unique representative of the form (y;z) and one has

c_X;z(y)c_X;z(y⁰)()(y⁰;z)(z;y)2G_‡()y⁰y:

Moreoverc_X;zis also equivariant since one has

c_X;z(u(y))ˆ(u(y);z)ˆ(u(y);u(z))(u(z);z)ˆc_X;z(y)g_x:

Let then~x2Xandy~2X⁰be two lifts ofxandy. The mapf :ˆc_X¹0;~yrc_X;~xis a non-decreasing map fromXtoX⁰and one has, usingr(gg_x)ˆr(g)g^k_y, that

f(u(x))ˆu^0k(f(x)); 8x2X:

One derives by construction the equality (f(a);~y)ˆr((a;~x)),8a2X. Takingaˆ~x this givesf(~x)ˆ~ysincer((~x;~x)) is a unit. This shows thatr(g)ˆG(f)(g)8g2Gxand the same equality holds for all g2G since bothr andG(f) are homomorphisms while any element ofGis of the formg(g⁰) ¹withg;g⁰2G_x. p REMARK3.10. The oriented groupoõÈds associated to archimedean sets are all equivalent, in the sense of equivalence of (small) categories, to the ordered group (Z;Z‡). It follows that a morphismfof oriented groupoõÈds induces an associated morphism Mod(f) of totally ordered groups, i.e. an ordered group morphism (Z;Z_‡)!(Z;Z_‡) given by multiplication by an integer Mod(f)ˆk2N. Propo- sition 3.9 suggests to refine the category gof oriented groupoõÈds by considering only the morphismsfsuch that Mod(f)6ˆ0. In other words what one requires is that the associated morphism of totally ordered groups, obtained by working modulo equivalence of categories, isinjective. One can then reformulate Proposi- tion 3.9 stating that the functorGis full and faithful.

COROLLARY 3.11. The epicyclic category L (cf. Appendix B) is canonically isomorphic to the category with objects the oriented groupoõÈds g(m), m0, of equation(14)and morphisms the non-trivial morphisms of oriented groupoõÈds.

The functor which associates to a morphism of oriented groupoõÈds its class up to equivalence coincides with the functorMod:L !N which sends a semilinear map of semimodules overFˆZ_maxto the corresponding injective endomorphism Fr_n2End(F)(cf.[5])

PROOF. By Proposition 2.8 of [5], the epicyclic categoryL is canonically iso- morphic to the full subcategory of Arc^j N whose objects are the archimedean sets m:ˆ(Z;x7!x‡m‡1) for m0. The oriented groupoõÈd G(m) is by Lemma 3.4 canonically isomorphic tog(m). The first statement then follows from Proposition 3.9 while the last one is checked easily and directly. p

4. Points of…L^op†^{^}and projective geometry in characteristic one

LetCbe a small category andC^the topos of contravariant functors fromCto the category of sets Gets. Yoneda's Lemma defines an embedding of the opposite categoryC^opinto the category of points of the toposC. More precisely, to an object^ c of C^op one associates the flat covariant functor h_c( ):C !Gets, h_c( )ˆ Hom_C(c; ). Then, one sees that through Yoneda's embeddingh:C^op !C,^ c7!h_c, any point ofC^can be obtained as a filtering colimit of points of the formhc. We shall apply these well known general facts toC ˆL^op: the opposite of the epicyclic cat- egoryL. We refer to Appendix Bfor the basic notations on the cyclic and epicyclic categories. It follows from Corollary 3.11 thatLis canonically isomorphic to the full subcategory of the categorygof oriented groupoõÈds whose objects are the oriented groupoõÈds of the formg(m). This fact suggests that one should obtain the points of the topos …L^op†^{^}by considering filtering colimits of the objectsg(m) ing. In this section we compare the colimit procedures taken respectively in the category of flat functorsL^op !Getsand in the categoryg. The comparison is made directly by reconstructing the structure of an oriented groupoõÈd starting from a flat functor as above. The main result is the following

THEOREM4.1. The category of points of the epicyclic topos…L^op†^{^}is equivalent to the categoryPwhose objects are pairs(K;E)where K is an algebraic extension of FˆZ_max and E is an archimedean semimodule over K. The morphisms are projective classes of semilinear maps and injective semifield morphisms.

One knows from [6] that an algebraic extensionKof the semifieldFˆZ_maxof tropical integers is equivalently described by a totally ordered group (H;H_‡) isomorphic to a subgroup ZHQ of the rationals. An archimedean semi- module EoverKis in turn described (cf.[5]) by a totally ordered setXon whichH acts by order automorphisms of type: (x;h)7!x‡hwhich fulfill the property

h‡x>x; 8h2H_‡; h6ˆ0; x2X …20†

and the archimedean condition

8x;y2X; 9h2H_‡ s:t: h‡x>y:

…21†

It follows from [1] (cf.also [10], Theorem 2 Chapter VII §5) that a point of a topos of the form C, where^ Cis a small category, is described by a covariant flat functor F:C !Gets. Next, we overview the strategy adopted to prove Theorem 4.1.

In §4.1 we associate to a pair (K;E) a point of …L^op†^{^}. This construction is accomplished in two steps. First, we extend the construction (X;u)7!G(X;u) of the oriented groupoõÈd associated to an archimedean set (as in §3.2) to a pair (K;E) as in Theorem 4.1. Then, for any given pair (K;E), we provide a natural con- struction of a point of …L^op†^{^} by means of the following associated flat functor (nˆ(Z;x7!x‡n‡1),n0)

F:L^op !Gets; F(n)ˆHom_g…g(n);G(K;E)†:

…22†

Here, one implements Corollary 3.11 to identify the categoryL with a full sub- category of the categorygof oriented groupoõÈds with injective morphisms (up to equivalence).

To produce the converse of the above construction,i.e.in order to show that any point of…L^op†^{^}is obtained as in (22) by means of a uniquely associated pair (K;E), we start from a covariant flat functorF:L^op !Gets and describe in

§§4.2-4.4 a procedure that allows one to reconstruct the semifieldKby using the natural geometric morphism of topoi associated to the functor Mod:L^op !N. The archimedean semimodule E (totally ordered set) is then reconstructed by using a suitable restriction ofFto obtain intervals from points of the simplicial toposD.^

4.1 ±The flat functorL^op !Getsassociated to a pair(K;E)

Let (H;H‡) be a totally ordered abelian group, denoted additively and X a totally ordered set on which the ordered groupHacts preserving the order and fulfilling (20). Let (G(X;H);G_‡(X;H)) be the oriented groupoõÈd associated to the pair (X;H) by Lemma 3.3, thus one has

G(X;H):ˆ(XX)=H; G‡(X;H):ˆ f(x;y)jxyg:

…23†

The next lemma is used to show that the functorF:L^op !Getsnaturally asso- ciated to a pair (K;E) isfiltering.

LEMMA4.2. Let(H;H‡)be a non-trivial subgroup of(Q;Q‡)and assume that the totally ordered set X on which H acts fulfills the archimedean condition(21).

LetF ˆ ff_j j1jngbe a finite set of morphismsf_j2Hom_g(g(m_j);G(X;H)).

Then, there exists a cyclic subgroup H₀H, a subset X₀X stable under the action of H0, morphismsc_j2Homg(g(mj);G(X0;H0))and an integer m2Nsuch

that, denoting byi:G(X0;H0)!G(X;H)the natural morphism, one has f_jˆic_j 8j; G(X₀;H₀)'g(m):

…24†

Moreover, let c;c⁰2Hom_g(g(n);G(X₀;H₀))be two morphisms such that icˆ ic⁰, then there exists a singly generated subgroup H₁with H₀H₁H, a subset X₁X containing X₀ and stable under the action of H₁, such that the equality i1cˆi1c⁰ holds inHomg(g(n);G(X1;H1)), withi1:G(X0;H0)!G(X1;H1)the natural morphism.

PROOF. We denote bya(m;i):ˆ(i;1)2…Z=(m‡1)Z†^j Zˆg(m) the natural positive generators of the oriented groupoõÈd g(m). Let (G;G_‡) be an oriented groupoõÈd. A morphism f2Homg(g(m);G) is uniquely specified by them‡1 ele- mentsg_iˆf(a(m;i))2G‡(cf.Figure 1) fulfilling the conditions (withg_m‡1:ˆg₀)

r(g_i)ˆs(g_i‡1); 8i;0im; g_m g₀2=G⁽⁰⁾:

IfGˆG(X;H), it follows that Homg(g(m);G) is the quotient of the subset ofX^m‡2 f(x₀;. . .;x_m‡1)2X^m‡2jx_j x_j‡1;8jn; x_m‡12x₀‡H_‡g

…25†

by the diagonal action of H. The morphism f2Hom_g(g(m);G) associated to (x0;. . .;xm‡1) is given by

f(a(m;i))ˆ(x_i‡1;x_i)2G‡ 8i; 0im:

…26†

For eachf_j2 Fwe get anh_j2H_‡, then we choose a finite subsetZ_jXsuch that f_jis represented by an…m‡2†-uple (x₀;. . .;x_m‡1)2X^m‡2with allx_i2Z_j. LetH₀ be the subgroup ofHgenerated by theh_j's and letX₀be theH₀invariant subset ofX generated by the union of theZj. ThenH0is singly generated and the pair (X0;H0)

Figure 1. One encodes a morphism fof oriented groupoõÈds fromg(n) tog(m) by the arrows g_iˆf(a(n;i)) associated to the generatorsa(n;i). To each generator (in blue) one assigns an arrow (in red) specified by its source and range and by an integer which gives the number of additional windings.

fulfills the archimedean condition (21). Moreover by construction one can lift the mapsf_j 2 F to elementsc_j 2Hom_g(g(m_j);G(X₀;H₀)) such thatf_j ˆic_j, where i:G(X₀;H₀)!G(X;H) is the natural morphism.

It remains to show that G(X₀;H₀)'g(m) for some m2N. By construction there exists a finite subsetZX0such thatX0ˆZ‡H0. Using the archimedean property (21) and sinceH0'Z, it follows that the pair (X0;H0) is an archimedean set such that the quotientX0=H0is finite. ThusG(X0;H0)'g(m), wherem‡1 is the cardinality ofX₀=H₀.

To prove the last statement, let (x₀;. . .;x_m‡1)2X₀^m‡2 (resp. (x⁰₀;. . .;x⁰_m‡1)2 X₀^m‡2) representc (resp.c⁰). The equality icˆic⁰ implies that there exists h2Hsuch thatx⁰_j ˆxj‡hfor allj. One then letsH1be the subgroup ofHgen- erated byH0 andhandX1theH1 invariant subset ofXgenerated byX0. p PROPOSITION4.3. Let(H;H_‡)be a non-trivial subgroup of(Q;Q_‡)and assume that the totally ordered set X on which H acts fulfills the archimedean condition (21). Then the following formula defines a flat functor

F:L^op !Gets F(n)ˆHomg(g(n);G); GˆG(X;H) …27†

where nˆ(Z;x 7!^u x‡n‡1).

PROOF. The statement follows from Lemma 4.2 showing that the functorFis obtained as a filtering colimit ofrepresentable, flatfunctors. We provide the detailed proof for completeness and to review the basic properties of flat functors which will be used later in this article. Corollary 3.11 provides a canonical identification of the epicyclic categoryLwith the full subcategory ofgof oriented groupoõÈds of the form g(m). In particular, (27) defines a covariant functor. It remains to show that this functor is flat. One knows from classical facts in the theory of Grothendieck topoi (cf. e.g.[10], Chapter VII §6, Theorem 3) that a functorF:C !Gets(Ca small category) is flat if and only if it is filteringi.e.the categoryR

C Fis filtering (cf.[10]

Chapter VII §6, Definition 2). The objects of the categoryR

C Fare pairs (j;x) wherej isan object ofCandx2F(j). The morphisms between two suchobjects (j;x) and (k;y) are elementsg2Hom_C(j;k) such thatF(g)xˆy. We recall that the filtering condition on a small categoryIis equivalent to the fulfillment of the following conditions

(1) Iis non empty.

(2) For any two objectsi;jofIthere exist an objectkand morphismsk!i, k!j.

(3) For any two morphisms a;b:i!j, there exist an object k and a morphismg:k!isuch thatagˆbg.

For each objectiof the small categoryCone obtains a flat functor provided by the Yoneda embeddingh_i:C !Gets; j7!Hom_C(i;j). Here we takeC ˆL^op and Fgiven by (27). The filtering property ofFonly involves finitely many elements of

Homg(g(n);G) and hence by Lemma 4.2, it follows using the filtering property of the functorsh_i. The first part of Lemma 4.2 is used to prove the filtering property (ii) while the last part is implemented to prove (iii). p

4.2 ±The image of a flat functor F:L^op !Getsby the module morphism The functor Mod:Arc^j N !N associates to any morphism of archime- dean sets the integerk2Ninvolved in Definition 3.8 (cf.[5]). Its restriction to the full subcategory whose objects are archimedean sets of type (Z;u), where u(x)ˆx‡n‡1, defines a functor Mod:L !N. The categoryNis isomorphic to its opposite in view of the commutativity of the multiplicative monoõÈd of positive integers. The functor Mod:L^op !Ndetermines a geometric morphism of topoi.

We recall once again ([6]) that the category of points of the toposNdis canonically equivalent to the category of totally ordered groups isomorphic to non-trivial subgroups of (Q;Q‡), and injective morphisms of ordered groups. This latter cat- egory is in turn equivalent to the category of algebraic extensions of the semifield FˆZ_max i.e. of extensions FK Q_max. The morphisms are the injective morphisms of semifields.

This section is devoted to the description of the action of the geometric morphism Mod:…L^op†^{^} !Ndon points, in terms of the associated flat functors.

This process allows one to recover the extensionK ofFˆZmax involved in The- orem 4.1 from the datum of a flat functorL^op !Gets.

Given two small categories C_j (jˆ1;2), a functor f:C₁ ! C₂ determines a geometric morphism (also noted f) of topoi C^₁ !C^₂ (cf. e.g. [10], Chapter VII §2, Theorem 2). The inverse image f sends an object of C^2, i.e. a con- travariant functor C2 !Gets to its composition with f which determines a contravariant functor C1 !Gets. The geometric morphism f sends points of C^₁ to points ofC^₂. In terms of the flat functors associated to points, the image by f of a flat functor F₁:C₁ !Gets associated to a point p₁:Gets !C^₁ is the flat functor F₂:C₂ !Gets obtained by composing the Yoneda embed- ding C2 !C^2 with p₁f, where p₁: ^C1 !Gets is the inverse image functor with respect to F1. Thus, for any object Z of C2 one obtains

F2(Z)ˆp₁(X); X:C^op₁ !Gets; X(c1):ˆHomC2(f(c1);Z):

…28†

We apply this procedure to the functor Mod:L^op !N: i.e.we take C₁ˆL^op, C₂ ˆNandfˆMod. LetFbe a flat functorF:L^op !Gets. The inverse image functor with respect toFof a covariant functorX:L !Getscoincides with the geometric realizationjXj_Fand it is of the form

jXj_F ˆ a

n0

(F(n)_LX(n))

!

=:

The image of the flat functorF:L^op !Getsby the morphism Mod is thus the flat functorH:N !Getsobtained as the geometric realizationjXj_Fof the covariant functor

X:L !Gets; X(n)ˆHomN(;Mod(n))N Obj(N)ˆ fg:

The functorXassociates to any object ofL the setNand to a morphismgofLits module Mod(g) acting by multiplication onN. Hence we obtain

H()ˆ a

n0

(F(n)_LN)

!

=:

…29†

The equivalence relation is exploited as follows: for (z;k)2F(n)N, one has (z;k)(F(g)z;k)2F(0)Nfor anyg2HomL(0;n), since Mod(g)ˆ1. Moreover for (z;k)2F(0)N, one has (z;k)(F(g)z;1) forg2Hom_L(0;0) with Mod(g)ˆk.

This shows that any element ofH() is equivalent to an element of the form (z;1) for somez2F(0). In particular one deduces thatH() is a quotient ofF(0).

LEMMA 4.4. Let F:L^op !Gets be a flat functor and (H;H‡) the corre- sponding point of dN through the morphism Mod. Then there is a canonical, surjectiveN-equivariant map

p:F(0)!H_‡; p(F(g)z)ˆMod(g)p(z); 8g2Hom_L(0;0):

Moreover, the equivalence relation xx⁰()p(x)ˆp(x⁰)is given by xx⁰() 9z2F(1); F(d0)zˆx; F(d1)zˆx⁰ …30†

whered_j 2Hom_D(0;1); (jˆ1;2)are the two face maps.

PROOF. Let p:F(0)!H(), p(z)ˆ(z;1) as in (29). As remarked above, the mapp is surjective. Let g2Hom_L(0;0), then one derives easily: p(F(g)z)ˆ (F(g)z;1)(z;Mod(g))ˆMod(g)p(z). Moreover, p(x)ˆp(x⁰) if and only if there exist n2N, u2F(n) and g;g⁰2Hom_L(0;n) such that F(g)uˆx, F(g⁰)uˆx⁰, Mod(g)ˆMod(g⁰):LetkˆMod(g), andC_k(n) the archimedean set obtained from nˆ(Z;u) by replacing u with u^k (cf.Appendix B). Then one derives canonical factorizations involving the identity map Z!Z viewed as the element Id^k_n2 Hom_L(Ck(n);n) with Mod(Id^k_n)ˆk

gˆId^k_na0; g⁰ˆId^k_na1; a0;a12HomL(0;C_k(n)):

Let dj2HomD(0;1), jˆ0;1, be the two face maps. One can then find a2HomL(1;C_k(n)) such that a_jˆad_j. Thus it follows that gˆId^k_nad0, g⁰ˆId^k_nad₁, and one also gets that xˆF(g)zˆF(d₀)F(a)F(Id^k_n)uˆF(d₀)z, zˆF(a)F(Id^k_n)u; x⁰ˆF(d₁)z;which proves (30). p